18 CANONICAL DECOMPOSITIONS OF A MEASURE 207 the existence, write \JL = /• p and v = g - p, where p is a positive measure and /^ 0, and let M be the p-measurable set of points x e X such thatf(x) > 0, so that \JL is concentrated on M (13.14.1). Then the measures v' = <pMg - p and v" = v — v' = (1 — <PM)# ' P satisfy the conditions of the theorem. For v" is disjoint from \JL by virtue of (13.18.1); also if A is a /z-negligible set, then A n M is p-negligible ((13.14.1) and (13.6,3)), hence A is v'-negligible (13.14.1) and consequently v' is a measure with base \JL (13.15.5). It is clear that if v is positive then so are v' and v" ; the formula v' = sup(inf(v, /7/z)) then n follows from the corresponding formula (pMg = sup(inf(#, «/)), which is n valid almost everywhere with respect to p. A measure /^ on X is said to be diffuse if n({x}) = 0 for all x E X. For example, Lebesgue measure on R is diffuse. With respect to a diffuse measure \JL every denumerable set is ^-negligible*, equivalently, a diffuse measure is concentrated on the complement of any denumerable set. On a denumer- able discrete space, the only diffuse measure is therefore n = 0. The sum of two diffuse measures is diffuse (13.16.1). The least upper bound of any set of diffuse positive measures which is bounded above is diffuse, by (13.15.9) and (13. 15.4). (13.18.5) If \JL is any measure on X, the set A of points xeX such that \I*\({x}) > 0 -is at most denumerable. Since X is the union of a sequence of compact sets Kn , it is enough to show that each of the sets A n Kn is at most denumerable. For this, it is enough to show that, for each integer m JS> 1 , the set Amn of points x e A n Kn such that M(M) ^ \l m infinite', now this is immediate, for if B c Amn consists of p points, then we have p/m£ \p\(B) ^ A measure /z on X is said to be atomic if it is concentrated on an at most denumerable set. From this definition and from (1 3.1 8.1 ) it follows immediately that an atomic measure and a diffuse measure are always disjoint. The sum of two atomic measures is atomic. The least upper bound of a set of atomic positive measures which is bounded above is atomic ((13.15.4) and (13.15.9)). (13.18.6) Every measure ^ can be expressed uniquely in the form ^ + /x2, where /^ is a diffuse measure and ju2 an atomic measure. The uniqueness of the decomposition follows from the fact that an atomic measure and a diffuse measure are disjoint. To establish the existence of the decomposition, it is enough to consider the at most denumerable set fr)l f°r ^1 ° e fri* ^i], then we have